How to increase glare strength of just emission map?
Since you are using over exposed Emit pass as a mask, there is no way to get proper pixel mix with rendered image. Even more because of border's pixels contaminated by background color (on antialiased edge). Also in your answer I can see edge in your render. Probably a bit better because of differences on Multiply values rather than mix type.

What I would try is to add Dilate-Erode node on "mask" to get rid of border pixels. This also brings all edges of mask highlighted because over exposed glare node. To compensate these appeared pixels I tried isolated only "glare" object color (can be done many ways, luminance is not probably the best way) and bring this part a bit darker to original, so pixels can be mixed more smoothly into final.





Note: I disabled Denoise node since produced worse result and was extremely slow. I also add Multiply after Glare, but your "before" is probably better you don't work with such low values, but result should be mostly the same. Environment texture was not attached so I used blender's Environment "studio.exr" texture.

These mixing of border pixels is always very sensitive for input, so its always quite a lot of tweaking. It's not like one group node you can attach anywhere.